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32,500 | In triangle $ABC$, $a=3$, $\angle C = \frac{2\pi}{3}$, and the area of $ABC$ is $\frac{3\sqrt{3}}{4}$. Find the lengths of sides $b$ and $c$. | \sqrt{13} | 1.5625 |
32,501 | Four rectangular strips each measuring $4$ by $16$ inches are laid out with two vertical strips crossing two horizontal strips forming a single polygon which looks like a tic-tack-toe pattern. What is the perimeter of this polygon?
[asy]
size(100);
draw((1,0)--(2,0)--(2,1)--(3,1)--(3,0)--(4,0)--(4,1)--(5,1)--(5,2)--(4,2)--(4,3)--(5,3)--(5,4)--(4,4)--(4,5)--(3,5)--(3,4)--(2,4)--(2,5)--(1,5)--(1,4)--(0,4)--(0,3)--(1,3)--(1,2)--(0,2)--(0,1)--(1,1)--(1,0));
draw((2,2)--(2,3)--(3,3)--(3,2)--cycle);
[/asy] | 80 | 26.5625 |
32,502 | A cowboy is 6 miles south of a stream which flows due east. He is also 12 miles west and 10 miles north of his cabin. Before returning to his cabin, he wishes to fill his water barrel from the stream and also collect firewood 5 miles downstream from the point directly opposite his starting point. Find the shortest distance (in miles) he can travel to accomplish all these tasks.
A) $11 + \sqrt{276}$
B) $11 + \sqrt{301}$
C) $11 + \sqrt{305}$
D) $12 + \sqrt{280}$ | 11 + \sqrt{305} | 9.375 |
32,503 | Solve the equations:
1. $2x^{2}+4x+1=0$ (using the method of completing the square)
2. $x^{2}+6x=5$ (using the formula method) | -3-\sqrt{14} | 3.90625 |
32,504 | Compute the number of two digit positive integers that are divisible by both of their digits. For example, $36$ is one of these two digit positive integers because it is divisible by both $3$ and $6$ .
*2021 CCA Math Bonanza Lightning Round #2.4* | 14 | 64.0625 |
32,505 | Points $A$ , $B$ , $C$ , $D$ , and $E$ are on the same plane such that $A,E,C$ lie on a line in that order, $B,E,D$ lie on a line in that order, $AE = 1$ , $BE = 4$ , $CE = 3$ , $DE = 2$ , and $\angle AEB = 60^\circ$ . Let $AB$ and $CD$ intersect at $P$ . The square of the area of quadrilateral $PAED$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?
*2021 CCA Math Bonanza Individual Round #9* | 967 | 7.03125 |
32,506 | At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability, then Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to be orange, the probability that there is a second orange marker in the hat can be expressed as simplified fraction $\tfrac{m}{n}$ . Find $m+n$ . | 39 | 4.6875 |
32,507 | Given point $F(0,1)$, moving point $M$ lies on the line $l:y=-1$. The line passing through point $M$ and perpendicular to the $x$-axis intersects the perpendicular bisector of segment $MF$ at point $P$. Let the locus of point $P$ be curve $C$.
$(1)$ Find the equation of curve $C$;
$(2)$ Given that the circle $x^{2}+(y+2)^{2}=4$ has a diameter $AB$, extending $AO$ and $BO$ intersect curve $C$ at points $S$ and $T$ respectively, find the minimum area of quadrilateral $ABST$. | 36 | 4.6875 |
32,508 | Determine the number of ways to arrange the letters of the word "PERCEPTION". | 907,200 | 0 |
32,509 | Given $2$ red and $2$ white balls, a total of $4$ balls are randomly arranged in a row. The probability that balls of the same color are adjacent to each other is $\_\_\_\_\_\_$. | \frac{1}{3} | 94.53125 |
32,510 | Given the function $f(x)=\sin \left( \frac {5\pi}{6}-2x\right)-2\sin \left(x- \frac {\pi}{4}\right)\cos \left(x+ \frac {3\pi}{4}\right).$
$(1)$ Find the minimum positive period and the intervals of monotonic increase for the function $f(x)$;
$(2)$ If $x_{0}\in\left[ \frac {\pi}{3}, \frac {7\pi}{12}\right]$ and $f(x_{0})= \frac {1}{3}$, find the value of $\cos 2x_{0}.$ | - \frac {2 \sqrt {6}+1}{6} | 0.78125 |
32,511 | On the edges \(AB\), \(BC\), and \(AD\) of the tetrahedron \(ABCD\), points \(K\), \(N\), and \(M\) are chosen, respectively, such that \(AK:KB = BN:NC = 2:1\) and \(AM:MD = 3:1\). Construct the section of the tetrahedron by the plane passing through points \(K\), \(M\), and \(N\). In what ratio does this plane divide edge \(CD\)? | 4:3 | 3.125 |
32,512 | The arithmetic sequence \( a, a+d, a+2d, a+3d, \ldots, a+(n-1)d \) has the following properties:
- When the first, third, fifth, and so on terms are added, up to and including the last term, the sum is 320.
- When the first, fourth, seventh, and so on, terms are added, up to and including the last term, the sum is 224.
What is the sum of the whole sequence? | 608 | 3.90625 |
32,513 | When Person A has traveled 100 meters, Person B has traveled 50 meters. When Person A reaches point $B$, Person B is still 100 meters away from $B$. Person A immediately turns around and heads back towards $A$, and they meet 60 meters from point $B$. What is the distance between points $A$ and $B$ in meters? | 300 | 1.5625 |
32,514 | In $\triangle ABC$, if $bc=3$, $a=2$, then the minimum value of the area of the circumcircle of $\triangle ABC$ is $\_\_\_\_\_\_$. | \frac{9\pi}{8} | 3.125 |
32,515 | Compute the number of ordered pairs of integers $(x,y)$ with $1\le x<y\le 50$ such that $i^x+i^y$ is a real number, and additionally, $x+y$ is divisible by $4$. | 288 | 2.34375 |
32,516 | The numbers 1, 2, ..., 2016 are written on a board. You are allowed to erase any two numbers and write their arithmetic mean instead. How must you proceed so that the number 1000 remains on the board? | 1000 | 56.25 |
32,517 | Given that the line $2mx+ny-4=0$ passes through the point of intersection of the function $y=\log _{a}(x-1)+2$ where $a>0$ and $a\neq 1$, find the minimum value of $\frac{1}{m}+\frac{4}{n}$. | 3+2\sqrt{2} | 44.53125 |
32,518 | A certain product has a cost price of $40$ yuan per unit. When the selling price is $60$ yuan per unit, 300 units can be sold per week. It is now necessary to reduce the price for clearance. According to market research, for every $1$ yuan reduction in price, an additional 20 units can be sold per week. Answer the following questions under the premise of ensuring profitability:
1. If the price reduction per unit is $x$ yuan and the profit from selling the goods per week is $y$ yuan, write the function relationship between $y$ and $x$, and determine the range of values for the independent variable $x$.
2. How much should the price be reduced by to maximize the profit per week? What is the maximum profit? | 6125 | 83.59375 |
32,519 | Let $L_1$ and $L_2$ be perpendicular lines, and let $F$ be a point at a distance $18$ from line $L_1$ and a distance $25$ from line $L_2$ . There are two distinct points, $P$ and $Q$ , that are each equidistant from $F$ , from line $L_1$ , and from line $L_2$ . Find the area of $\triangle{FPQ}$ . | 210 | 12.5 |
32,520 | Five cards with different numbers are given: $-5$, $-4$, $0$, $+4$, $+6$. Two cards are drawn from them. The smallest quotient obtained by dividing the numbers on these two cards is ____. | -\dfrac{3}{2} | 1.5625 |
32,521 | Given the lines $l_{1}$: $x+ay-a+2=0$ and $l_{2}$: $2ax+(a+3)y+a-5=0$.
$(1)$ When $a=1$, find the coordinates of the intersection point of lines $l_{1}$ and $l_{2}$.
$(2)$ If $l_{1}$ is parallel to $l_{2}$, find the value of $a$. | a = \frac{3}{2} | 36.71875 |
32,522 | In the game of *Winners Make Zeros*, a pair of positive integers $(m,n)$ is written on a sheet of paper. Then the game begins, as the players make the following legal moves:
- If $m\geq n$ , the player choose a positive integer $c$ such that $m-cn\geq 0$ , and replaces $(m,n)$ with $(m-cn,n)$ .
- If $m<n$ , the player choose a positive integer $c$ such that $n-cm\geq 0$ , and replaces $(m,n)$ with $(m,n-cm)$ .
When $m$ or $n$ becomes $0$ , the game ends, and the last player to have moved is declared the winner. If $m$ and $n$ are originally $2007777$ and $2007$ , find the largest choice the first player can make for $c$ (on his first move) such that the first player has a winning strategy after that first move. | 999 | 8.59375 |
32,523 | Given the numbers $12534, 25341, 53412, 34125$, calculate their sum. | 125412 | 99.21875 |
32,524 | Let $a < b < c < d < e$ be real numbers. We calculate all possible sums in pairs of these 5 numbers. Of these 10 sums, the three smaller ones are 32, 36, 37, while the two larger ones are 48 and 51. Determine all possible values that $e$ can take. | 27.5 | 13.28125 |
32,525 | In a mathematics competition conducted at a school, the scores $X$ of all participating students approximately follow the normal distribution $N(70, 100)$. It is known that there are 16 students with scores of 90 and above (inclusive of 90).
(1) What is the approximate total number of students who participated in the competition?
(2) If the school plans to reward students who scored 80 and above (inclusive of 80), how many students are expected to receive a reward in this competition?
Note: $P(|X-\mu| < \sigma)=0.683$, $P(|X-\mu| < 2\sigma)=0.954$, $P(|X-\mu| < 3\sigma)=0.997$. | 110 | 28.90625 |
32,526 | Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction. | \frac{1}{8} | 7.03125 |
32,527 | In the diagram below, $AB = 30$ and $\angle ADB = 90^\circ$. If $\sin A = \frac{3}{5}$ and $\sin C = \frac{1}{4}$, what is the length of $DC$? | 18\sqrt{15} | 25 |
32,528 | A point $P(\frac{\pi}{12}, m)$ on the graph of the function $y = \sin 2x$ can be obtained by shifting a point $Q$ on the graph of the function $y = \cos (2x - \frac{\pi}{4})$ to the left by $n (n > 0)$ units. Determine the minimum value of $n \cdot m$. | \frac{5\pi}{48} | 71.875 |
32,529 | Given a list of $3000$ positive integers with a unique mode occurring exactly $12$ times, calculate the least number of distinct values that can occur in the list. | 273 | 48.4375 |
32,530 | Given an ellipse $C$ with its center at the origin and its foci on the $x$-axis, and its eccentricity equal to $\frac{1}{2}$. One of its vertices is exactly the focus of the parabola $x^{2}=8\sqrt{3}y$.
(Ⅰ) Find the standard equation of the ellipse $C$;
(Ⅱ) If the line $x=-2$ intersects the ellipse at points $P$ and $Q$, and $A$, $B$ are points on the ellipse located on either side of the line $x=-2$.
(i) If the slope of line $AB$ is $\frac{1}{2}$, find the maximum area of the quadrilateral $APBQ$;
(ii) When the points $A$, $B$ satisfy $\angle APQ = \angle BPQ$, does the slope of line $AB$ have a fixed value? Please explain your reasoning. | \frac{1}{2} | 10.15625 |
32,531 | Given the line $y=kx+b\left(k \gt 0\right)$ is tangent to the circle $x^{2}+y^{2}=1$ and the circle $\left(x-4\right)^{2}+y^{2}=1$, find $k=$____ and $b=$____. | -\frac{2\sqrt{3}}{3} | 1.5625 |
32,532 | A point is chosen at random on the number line between 0 and 1, and the point is colored red. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored blue. What is the probability that the number of the blue point is greater than the number of the red point, but less than three times the number of the red point? | \frac{1}{9} | 7.8125 |
32,533 | Juan rolls a fair regular octahedral die marked with the numbers 1 through 8. Amal now rolls a fair twelve-sided die marked with the numbers 1 through 12. What is the probability that the product of the two rolls is a multiple of 4? | \frac{7}{16} | 15.625 |
32,534 | In Sichuan, the new college entrance examination was launched in 2022, and the first new college entrance examination will be implemented in 2025. The new college entrance examination adopts the "$3+1+2$" mode. "$3$" refers to the three national unified examination subjects of Chinese, Mathematics, and Foreign Languages, regardless of whether they are arts or sciences; "$1$" refers to choosing one subject from Physics and History; "$2$" refers to choosing two subjects from Political Science, Geography, Chemistry, and Biology. The subject selection situation of the first-year high school students in a certain school in 2022 is shown in the table below:
| Subject Combination | Physics, Chemistry, Biology | Physics, Chemistry, Political Science | Physics, Chemistry, Geography | History, Political Science, Geography | History, Political Science, Biology | History, Chemistry, Political Science | Total |
|---------------------|-----------------------------|-------------------------------------|-----------------------------|--------------------------------------|----------------------------------|--------------------------------------|-------|
| Male | 180 | 80 | 40 | 90 | 30 | 20 | 440 |
| Female | 150 | 70 | 60 | 120 | 40 | 20 | 460 |
| Total | 330 | 150 | 100 | 210 | 70 | 40 | 900 |
$(1)$ Complete the $2\times 2$ contingency table below and determine if there is a $99\%$ certainty that "choosing Physics is related to the gender of the student":
| | Choose Physics | Do not choose Physics | Total |
|-------------------|----------------|-----------------------|-------|
| Male | | | |
| Female | | | |
| Total | | | |
$(2)$ From the female students who chose the combinations of History, Political Science, Biology and History, Chemistry, Political Science, select 6 students using stratified sampling to participate in a history knowledge competition. Find the probability that the 2 selected female students are from the same combination.
Given table and formula: ${K^2}=\frac{{n{{(ad-bc)}^2}}}{{(a+b)(c+d)(a+c)(b+d)}}$
| $P(K^{2}\geqslant k_{0})$ | 0.15 | 0.1 | 0.05 | 0.01 |
|---------------------------|------|------|------|------|
| $k_{0}$ | 2.072| 2.706| 3.841| 6.635| | \frac{7}{15} | 34.375 |
32,535 | Suppose the estimated €25 billion (Euros) cost to send a person to the planet Mars is shared equally by the 300 million people in a consortium of countries. Given the exchange rate of 1 Euro = 1.2 dollars, calculate each person's share in dollars. | 100 | 86.71875 |
32,536 | Given that Mike walks to his college, averaging 70 steps per minute with each step being 80 cm long, and it takes him 20 minutes to get there, determine how long it takes Tom to reach the college, given that he averages 120 steps per minute, but his steps are only 50 cm long. | 18.67 | 79.6875 |
32,537 | Let \( a \) and \( b \) be positive integers such that \( 15a + 16b \) and \( 16a - 15b \) are both perfect squares. Find the smallest possible value among these squares. | 481^2 | 0 |
32,538 | Given the function $$f(x)=2\sin(wx+\varphi+ \frac {\pi}{3})+1$$ where $|\varphi|< \frac {\pi}{2}$ and $w>0$, is an even function, and the distance between two adjacent axes of symmetry of the function $f(x)$ is $$\frac {\pi}{2}$$.
(1) Find the value of $$f( \frac {\pi}{8})$$.
(2) When $x\in(-\frac {\pi}{2}, \frac {3\pi}{2})$, find the sum of the real roots of the equation $f(x)= \frac {5}{4}$. | 2\pi | 69.53125 |
32,539 | The positive integer divisors of 252, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 14? | 70 | 1.5625 |
32,540 | In the diagram \(PQRS\) is a rhombus. Point \(T\) is the midpoint of \(PS\) and point \(W\) is the midpoint of \(SR\). What is the ratio of the unshaded area to the shaded area? | 1:1 | 1.5625 |
32,541 | Given the inequality $\ln (x+1)-(a+2)x\leqslant b-2$ that always holds, find the minimum value of $\frac {b-3}{a+2}$. | 1-e | 39.0625 |
32,542 | Select 5 elements from the set $\{x|1\leq x \leq 11, \text{ and } x \in \mathbb{N}^*\}$ to form a subset of this set, and any two elements in this subset do not sum up to 12. How many different subsets like this are there? (Answer with a number). | 112 | 3.125 |
32,543 | Given that Jennifer plans to build a fence around her garden in the shape of a rectangle, with $24$ fence posts, and evenly distributing the remaining along the edges, with $6$ yards between each post, and with the longer side of the garden, including corners, having three times as many posts as the shorter side, calculate the area, in square yards, of Jennifer’s garden. | 855 | 0 |
32,544 | What is the area, in square units, of a trapezoid bounded by the lines $y = x$, $y = 15$, $y = 5$ and the line $x = 5$? | 50 | 50.78125 |
32,545 | The least common multiple of $x$ and $y$ is $18$, and the least common multiple of $y$ and $z$ is $20$. Determine the least possible value of the least common multiple of $x$ and $z$. | 90 | 24.21875 |
32,546 | Given the parabola $C$: $y^2=2px (p > 0)$ with focus $F$ and directrix $l$. A line perpendicular to $l$ at point $A$ on the parabola $C$ at $A(4,y_0)$ intersects $l$ at $A_1$. If $\angle A_1AF=\frac{2\pi}{3}$, determine the value of $p$. | 24 | 0.78125 |
32,547 | Athletes A, B, and C, along with 4 volunteers, are to be arranged in a line. If A and B are next to each other and C is not at either end, determine the number of different ways to arrange them. | 960 | 27.34375 |
32,548 | Seventy percent of a train's passengers are men, and fifteen percent of those men are in the business class. What is the number of men in the business class if the train is carrying 300 passengers? | 32 | 83.59375 |
32,549 | Calculate the monotonic intervals of $F(x)=\int_{0}^{x}{(t^{2}+2t-8)dt}$ for $x > 0$.
(1) Determine the monotonic intervals of $F(x)$.
(2) Find the maximum and minimum values of the function $F(x)$ on the interval $[1,2]$. | -\frac{28}{3} | 44.53125 |
32,550 | In the spring college entrance examination of Shanghai in 2011, there were 8 universities enrolling students. If exactly 3 students were admitted by 2 of these universities, the number of ways this could happen is ____. | 168 | 3.125 |
32,551 | Consider the numbers $\{24,27,55,64,x\}$ . Given that the mean of these five numbers is prime and the median is a multiple of $3$ , compute the sum of all possible positive integral values of $x$ . | 60 | 6.25 |
32,552 | Given triangle $ABC$, $\overrightarrow{CA}•\overrightarrow{CB}=1$, the area of the triangle is $S=\frac{1}{2}$,<br/>$(1)$ Find the value of angle $C$;<br/>$(2)$ If $\sin A\cos A=\frac{{\sqrt{3}}}{4}$, $a=2$, find $c$. | \frac{2\sqrt{6}}{3} | 63.28125 |
32,553 | A set of sample data $11$, $13$, $15$, $a$, $19$ has an average of $15$. Calculate the standard deviation of this data set. | 2\sqrt{2} | 83.59375 |
32,554 | Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\pi$. | \frac{9\pi}{2} - 9 | 78.125 |
32,555 | Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$, and it satisfies $S_{2014} > 0$, $S_{2015} < 0$. For any positive integer $n$, it holds that $|a_n| \geqslant |a_k|$, determine the value of $k$. | 1008 | 17.1875 |
32,556 | Given that \( PQ = 4 \), \( QR = 8 \), \( RS = 8 \), and \( ST = 3 \), if \( PQ \) is perpendicular to \( QR \), \( QR \) is perpendicular to \( RS \), and \( RS \) is perpendicular to \( ST \), calculate the distance from \( P \) to \( T \). | 13 | 10.15625 |
32,557 | Sasa wants to make a pair of butterfly wings for her Barbie doll. As shown in the picture, she first drew a trapezoid and then drew two diagonals, which divided the trapezoid into four triangles. She cut off the top and bottom triangles, and the remaining two triangles are exactly a pair of beautiful wings. If the areas of the two triangles that she cut off are 4 square centimeters and 9 square centimeters respectively, then the area of the wings that Sasa made is $\qquad$ square centimeters. | 12 | 14.84375 |
32,558 | David, Delong, and Justin each showed up to a problem writing session at a random time during the session. If David arrived before Delong, what is the probability that he also arrived before Justin? | \frac{2}{3} | 10.9375 |
32,559 | Define $p(n)$ to be th product of all non-zero digits of $n$ . For instance $p(5)=5$ , $p(27)=14$ , $p(101)=1$ and so on. Find the greatest prime divisor of the following expression:
\[p(1)+p(2)+p(3)+...+p(999).\] | 103 | 86.71875 |
32,560 | Let \( a, b, c, d \) be integers such that \( a > b > c > d \geq -2021 \) and
\[ \frac{a+b}{b+c} = \frac{c+d}{d+a} \]
(and \( b+c \neq 0 \neq d+a \)). What is the maximum possible value of \( a \cdot c \)? | 510050 | 0 |
32,561 | You are given a positive integer $k$ and not necessarily distinct positive integers $a_1, a_2 , a_3 , \ldots,
a_k$ . It turned out that for any coloring of all positive integers from $1$ to $2021$ in one of the $k$ colors so that there are exactly $a_1$ numbers of the first color, $a_2$ numbers of the second color, $\ldots$ , and $a_k$ numbers of the $k$ -th color, there is always a number $x \in \{1, 2, \ldots, 2021\}$ , such that the total number of numbers colored in the same color as $x$ is exactly $x$ . What are the possible values of $k$ ?
*Proposed by Arsenii Nikolaiev* | 2021 | 25 |
32,562 | Calculate the sum of $2367 + 3672 + 6723 + 7236$. | 19998 | 38.28125 |
32,563 | Given the function $f(x)=\sin (2x+ \frac {π}{3})$, it is shifted right by $\frac {2π}{3}$ units, and then the resulting function's graph has each point's horizontal coordinate changed to twice its original value while the vertical coordinate remains unchanged, yielding the function $y=g(x)$. Calculate the area enclosed by the function $y=g(x)$, $x=- \frac {π}{2}$, $x= \frac {π}{3}$, and the $x$-axis. | \frac{3}{2} | 9.375 |
32,564 | The sum of the squares of four consecutive positive integers is 9340. What is the sum of the cubes of these four integers? | 457064 | 0 |
32,565 | The area of the base of a hemisphere is $144\pi$. The hemisphere is mounted on top of a cylinder that has the same radius as the hemisphere and a height of 10. What is the total surface area of the combined solid? Express your answer in terms of $\pi$. | 672\pi | 54.6875 |
32,566 | Given the parabola $y^2 = 2px (0 < p < 4)$, with a focus at point $F$, and a point $P$ moving along $C$. Let $A(4,0)$ and $B(p, \sqrt{2}p)$ with the minimum value of $|PA|$ being $\sqrt{15}$, find the value of $|BF|$. | \dfrac{9}{2} | 63.28125 |
32,567 | Given the function f(x) = 2cos^2(x) - 2$\sqrt{3}$sin(x)cos(x).
(I) Find the monotonically decreasing interval of the function f(x);
(II) Find the sum of all the real roots of the equation f(x) = $- \frac{1}{3}$ in the interval [0, $\frac{\pi}{2}$]. | \frac{2\pi}{3} | 13.28125 |
32,568 | What are the first three digits to the right of the decimal point in the decimal representation of $(10^{100} + 1)^{5/3}$? | 666 | 7.03125 |
32,569 | For a natural number \( N \), if at least five of the natural numbers from 1 to 9 can divide \( N \), then \( N \) is called a "five-rule number." What is the smallest "five-rule number" greater than 2000? | 2004 | 97.65625 |
32,570 | Given that $40\%$ of the birds were pigeons, $20\%$ were sparrows, $15\%$ were crows, and the remaining were parakeets, calculate the percent of the birds that were not sparrows and were crows. | 18.75\% | 51.5625 |
32,571 | Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), and $P$ is a point on the ellipse, with $\overrightarrow{PF_{1}} \cdot (\overrightarrow{OF_{1}} + \overrightarrow{OP}) = 0$, if $|\overrightarrow{PF_{1}}| = \sqrt{2}|\overrightarrow{PF_{2}}|$, determine the eccentricity of the ellipse. | \sqrt{6} - \sqrt{3} | 4.6875 |
32,572 | Circles $P$, $Q$, and $R$ are externally tangent to each other and internally tangent to circle $S$. Circles $Q$ and $R$ are congruent. Circle $P$ has radius 2 and passes through the center of $S$. What is the radius of circle $Q$? | \frac{16}{9} | 0.78125 |
32,573 | The diagram shows a square \(PQRS\) with edges of length 1, and four arcs, each of which is a quarter of a circle. Arc \(TRU\) has center \(P\); arc \(VPW\) has center \(R\); arc \(UV\) has center \(S\); and arc \(WT\) has center \(Q\). What is the length of the perimeter of the shaded region?
A) 6
B) \((2 \sqrt{2} - 1) \pi\)
C) \(\left(\sqrt{2} - \frac{1}{2}\right) \pi\)
D) 2 \(\pi\)
E) \((3 \sqrt{2} - 2) \pi\) | (2\sqrt{2} - 1)\pi | 0 |
32,574 | Two distinct numbers a and b are chosen randomly from the set $\{3, 3^2, 3^3, ..., 3^{20}\}$. What is the probability that $\mathrm{log}_a b$ is an integer?
A) $\frac{12}{19}$
B) $\frac{1}{4}$
C) $\frac{24}{95}$
D) $\frac{1}{5}$ | \frac{24}{95} | 72.65625 |
32,575 | Let $ABCDEF$ be a regular hexagon, and let $G$ , $H$ , $I$ , $J$ , $K$ , and $L$ be the midpoints of sides $AB$ , $BC$ , $CD$ , $DE$ , $EF$ , and $FA$ , respectively. The intersection of lines $\overline{AH}$ , $\overline{BI}$ , $\overline{CJ}$ , $\overline{DK}$ , $\overline{EL}$ , and $\overline{FG}$ bound a smaller regular hexagon. Find the ratio of the area of the smaller hexagon to the area of $ABCDEF$ . | 4/7 | 0 |
32,576 | The Aces are playing the Kings in a playoff series, where the first team to win 5 games wins the series. Each game's outcome leads to the Aces winning with a probability of $\dfrac{7}{10}$, and there are no draws. Calculate the probability that the Aces win the series. | 90\% | 0 |
32,577 | On a circle, points $A, B, C, D, E, F, G$ are located clockwise as shown in the diagram. It is known that $AE$ is the diameter of the circle. Additionally, it is known that $\angle ABF = 81^\circ$ and $\angle EDG = 76^\circ$. How many degrees is the angle $FCG$? | 67 | 7.8125 |
32,578 | For a function $f(x)$ with domain $I$, if there exists an interval $\left[m,n\right]\subseteq I$ such that $f(x)$ is a monotonic function on the interval $\left[m,n\right]$, and the range of the function $y=f(x)$ for $x\in \left[m,n\right]$ is $\left[m,n\right]$, then the interval $\left[m,n\right]$ is called a "beautiful interval" of the function $f(x)$;
$(1)$ Determine whether the functions $y=x^{2}$ ($x\in R$) and $y=3-\frac{4}{x}$ ($x \gt 0$) have a "beautiful interval". If they exist, write down one "beautiful interval" that satisfies the condition. (Provide the conclusion directly without requiring a proof)
$(2)$ If $\left[m,n\right]$ is a "beautiful interval" of the function $f(x)=\frac{{({{a^2}+a})x-1}}{{{a^2}x}}$ ($a\neq 0$), find the maximum value of $n-m$. | \frac{2\sqrt{3}}{3} | 7.03125 |
32,579 | Three people, A, B, and C, visit three tourist spots, with each person visiting only one spot. Let event $A$ be "the three people visit different spots," and event $B$ be "person A visits a spot alone." Then, the probability $P(A|B)=$ ______. | \dfrac{1}{2} | 58.59375 |
32,580 | For all positive integers $n$ , let $f(n)$ return the smallest positive integer $k$ for which $\tfrac{n}{k}$ is not an integer. For example, $f(6) = 4$ because $1$ , $2$ , and $3$ all divide $6$ but $4$ does not. Determine the largest possible value of $f(n)$ as $n$ ranges over the set $\{1,2,\ldots, 3000\}$ . | 11 | 34.375 |
32,581 | In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a\cos B - b\cos A = c$, and $C = \frac{π}{5}$, calculate the measure of $\angle B$. | \frac{3\pi}{10} | 87.5 |
32,582 | An object in the plane moves from the origin and takes a ten-step path, where at each step the object may move one unit to the right, one unit to the left, one unit up, or one unit down. How many different points could be the final point? | 221 | 14.84375 |
32,583 | BdMO National 2016 Higher Secondary
<u>**Problem 4:**</u>
Consider the set of integers $ \left \{ 1, 2, ......... , 100 \right \} $ . Let $ \left \{ x_1, x_2, ......... , x_{100} \right \}$ be some arbitrary arrangement of the integers $ \left \{ 1, 2, ......... , 100 \right \}$ , where all of the $x_i$ are different. Find the smallest possible value of the sum, $S = \left | x_2 - x_1 \right | + \left | x_3 - x_2 \right | + ................+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right | $ . | 198 | 72.65625 |
32,584 | Sarah baked 4 dozen pies for a community fair. Out of these pies:
- One-third contained chocolate,
- One-half contained marshmallows,
- Three-fourths contained cayenne pepper,
- One-eighth contained walnuts.
What is the largest possible number of pies that had none of these ingredients? | 12 | 10.15625 |
32,585 | Lynne chooses four distinct digits from 1 to 9 and arranges them to form the 24 possible four-digit numbers. These 24 numbers are added together giving the result \(N\). For all possible choices of the four distinct digits, what is the largest sum of the distinct prime factors of \(N\)? | 146 | 3.125 |
32,586 | Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ | 2028 | 44.53125 |
32,587 | A regular dodecagon \( Q_1 Q_2 \dotsb Q_{12} \) is drawn in the coordinate plane with \( Q_1 \) at \( (4,0) \) and \( Q_7 \) at \( (2,0) \). If \( Q_n \) is the point \( (x_n,y_n) \), compute the numerical value of the product
\[
(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{12} + y_{12} i).
\] | 531440 | 13.28125 |
32,588 | A line connects points $(2,1)$ and $(7,3)$ on a square that has vertices at $(2,1)$, $(7,1)$, $(7,6)$, and $(2,6)$. What fraction of the area of the square is above this line? | \frac{4}{5} | 9.375 |
32,589 | Compute $1-2+3-4+\dots+100-101$. | -151 | 30.46875 |
32,590 | Given the curve $C:\begin{cases}x=2\cos a \\ y= \sqrt{3}\sin a\end{cases} (a$ is the parameter) and the fixed point $A(0,\sqrt{3})$, ${F}_1,{F}_2$ are the left and right foci of this curve, respectively. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established.
$(1)$ Find the polar equation of the line $AF_{2}$;
$(2)$ A line passing through point ${F}_1$ and perpendicular to the line $AF_{2}$ intersects this conic curve at points $M$, $N$, find the value of $||MF_{1}|-|NF_{1}||$. | \dfrac{12\sqrt{3}}{13} | 1.5625 |
32,591 | Given the arithmetic sequence $\{a_n\}$, it is given that $a_2+a_8-a_{12}=0$ and $a_{14}-a_4=2$. Let $s_n=a_1+a_2+\ldots+a_n$, then determine the value of $s_{15}$. | 30 | 99.21875 |
32,592 | Maria is 54 inches tall, and Samuel is 72 inches tall. Using the conversion 1 inch = 2.54 cm, how tall is each person in centimeters? Additionally, what is the difference in their heights in centimeters? | 45.72 | 32.8125 |
32,593 | In triangle \( \triangle ABC \), angle \( \angle C \) is a right angle, \( AC = 3 \) and \( BC = 4 \). In triangle \( \triangle ABD \), angle \( \angle A \) is a right angle, and \( AD = 12 \). Points \( C \) and \( D \) are on opposite sides of \( AB \). A line passing through point \( D \) and parallel to \( AC \) intersects \( CB \) at \( E \). Given that \( \frac{DE}{DB} = \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers, what is \( m + n \)?
(42nd United States of America Mathematical Olympiad, 1991) | 128 | 0 |
32,594 | Find the value of $m$ for which the quadratic equation $3x^2 + mx + 36 = 0$ has exactly one solution in $x$. | 12\sqrt{3} | 48.4375 |
32,595 | Given angles $α$ and $β$ whose vertices are at the origin of coordinates, and their initial sides coincide with the positive half-axis of $x$, $α$, $β$ $\in(0,\pi)$, the terminal side of angle $β$ intersects the unit circle at a point whose x-coordinate is $- \dfrac{5}{13}$, and the terminal side of angle $α+β$ intersects the unit circle at a point whose y-coordinate is $ \dfrac{3}{5}$, then $\cos α=$ ______. | \dfrac{56}{65} | 13.28125 |
32,596 | Two teachers, A and B, and four students stand in a row. (Explain the process, list the expressions, and calculate the results, expressing the results in numbers)<br/>$(1)$ The two teachers cannot be adjacent. How many ways are there to arrange them?<br/>$(2)$ A is to the left of B. How many ways are there to arrange them?<br/>$(3)$ A must be at the far left or B must be at the far left, and A cannot be at the far right. How many ways are there to arrange them?<br/>$(4)$ The two teachers are in the middle, with two students at each end. If the students are of different heights and must be arranged from tallest to shortest from the middle to the ends, how many ways are there to arrange them? | 12 | 24.21875 |
32,597 | Given 6 teachers who will be allocated to two classes, where the maximum number of teachers in each class is 4, determine the number of different arrangements. | 50 | 21.875 |
32,598 | Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$ . How many passcodes satisfy these conditions? | 36 | 96.09375 |
32,599 | Consider numbers of the form \(10n + 1\), where \(n\) is a positive integer. We shall call such a number 'grime' if it cannot be expressed as the product of two smaller numbers, possibly equal, both of which are of the form \(10k + 1\), where \(k\) is a positive integer. How many 'grime numbers' are there in the sequence \(11, 21, 31, 41, \ldots, 981, 991\)?
A) 0
B) 8
C) 87
D) 92
E) 99 | 87 | 85.15625 |
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